ar X iv :h ep -t h/ 02 02 13 5v 2 2 8 A pr 2 00 2 NEIP-02-002 hep–th/0202135 An eight–dimensional approach to G2 manifolds Rafael Hernández1 and Konstadinos Sfetsos2 1 Institut de Physique, Université de Neuchâtel Breguet 1, CH-2000 Neuchâtel, Switzerland rafael.hernandez@unine.ch 2 Department of Engineering Sciences, University of Patras 26110 Patras, Greece sfetsos@mail.cern.ch, des.upatras.gr Abstract We develop a systematic approach to G2 holonomy manifolds with an SU(2) × SU(2) isometry using maximal eight-dimensional gauged supergravity to describe D6-branes wrapped on deformed three-spheres. A quite general metric ansatz that generalizes the celebrated Bryant–Salamon metric involves nine functions. We show that only six of them are the independent ones and derive the general first order system of differential equations that they obey. As a byproduct of our analysis, we generalize the notion of the twist that relates the spin and gauge connections in a way that involves non-trivially the scalar fields. http://arxiv.org/abs/hep-th/0202135v2 http://arxiv.org/abs/hep--th/0202135 Compactifications of M-theory on manifolds of exceptional holonomy have recently attracted great attention, mostly as a consequence of their relation to minimally super- symmetric gauge theories. Four dimensional N = 1 supersymmetry (in Minkowski space) requires the internal seven manifold to have G2 holonomy. But G2 holonomy also appears in the geometric dual description of the large N limit of four dimensional gauge theo- ries with four supercharges: the conjectured duality between D6-branes on the deformed conifold and a type IIA geometry with RR flux on the resolved conifold in [1] was bet- ter understood in terms of M-theory on a seven manifold of G2 holonomy [2], where it corresponds to a flop transition [3]. Extensions of this duality and construction of new metrics from diverse approaches have revived the study of compactifications on manifolds of exceptional holonomy [4]-[31]. The number of known complete metrics of G2 holonomy is still quite reduced. It is therefore of great interest to obtain new metrics of G2 holonomy in order to improve our understanding of the above dualities and compactifications. The aim of this letter is to elaborate on a gauged supergravity approach to the systematic construction of manifolds of G2 holonomy. Branes wrapped on supersymmetric cycles have also been lately quite extensively stud- ied within the framework of gauged supergravity as a promising candidate to gravity duals of field theories with low supersymmetry [32]-[48]. In [37] a configuration of D6-branes wrapping special Lagrangian 3-spheres was considered as a gravity dual of four dimen- sional field theories with N = 1 supersymmetry. The lift to eleven dimensions of the eight dimensional solution describing the deformation on the worldvolume of the wrapped branes was there shown to correspond to one of the known metrics of G2 holonomy [49]. In this letter we will show how gauged supergravity in eight dimensions provides a natural framework to construct general metrics of G2 holonomy by allowing deformations on the 3-cycle. We will derive the conditions to guarantee G2 holonomy on a seven manifold metric of the form ds2 7 = dr2 + 3 ∑ i=1 a2 iσ 2 i + 3 ∑ i=1 b2i (Σi + ciσi) 2 , where, as it will become clear from our analysis below, only six of the nine functions involved in this general metric are independent. In what follows we will briefly review some relevant facts about eight dimensional super- gravity. We will then construct the equations describing a supersymmetric configuration 1 corresponding to a set of D6-branes wrapped on a deformed 3-sphere. The lift to eleven dimensions of this configuration will prove to be a seven manifold of G2 holonomy with SU(2) × SU(2) isometry which includes some of the proposed ansatzs in the literature. Maximal gauged supergravity in eight dimensions was constructed by Salam and Sezgin [50] through Scherk–Schwarz compactification of eleven dimensional supergravity on an SU(2) group manifold. The field content in the gravity sector of the theory consists of the metric gµν , a dilaton Φ, five scalars given by a unimodular 3 × 3 matrix Li α in the coset SL(3,R)/SO(3) and an SU(2) gauge potential Aµ.1 In addition, on the fermion side we have the pseudo–Majorana spinors ψµ and χi. The Lagrangian density for the bosonic fields is given, in κ = 1 units, by L = 1 4 R− 1 4 e2ΦF α µνF µν βgαβ − 1 4 Pµ ijP µ ij − 1 2 (∂µΦ)2 − g2 16 e−2Φ(TijT ij − 1 2 T 2) , (1) where F α µν is the Yang–Mills field strength. Supersymmetry is preserved by bosonic solutions to the equations of motion of eight dimensional supergravity if the supersymmetry variations for the gaugino and the gravitino vanish. These are, respectively, given by δχi = 1 2 (Pµ ij + 2 3 δij∂µΦ)Γ̂jΓµǫ− 1 4 eΦFµν iΓ µνǫ− g 8 e−Φ(Tij − 1 2 δijT )ǫjklΓ̂klǫ = 0 (2) and δψγ = Dγǫ+ 1 24 eΦF i µν Γ̂i(Γ µν γ − 10δ µ γ Γν)ǫ− g 288 e−ΦǫijkΓ̂ ijkΓγTǫ = 0 . (3) The covariant derivative is Dµǫ = ∂µǫ+ 1 4 ωab µ Γabǫ+ 1 4 Qµ ijΓ̂ ijǫ , (4) where Pµ ij andQµ ij are, respectively, the symmetric and antisymmetric quantities entering the Cartan decomposition of the SL(3,R)/SO(3) coset, defined through Pµ ij +Qµ ij ≡ Lα i (∂µδ β α − g ǫαβγA γ µ)Lβ j , (5) and Tij is the T -tensor defining the potential energy associated to the scalar fields, T ij ≡ Li αL j βδ αβ , (6) 1The fields arising from reduction of the eleven dimensional three-form are a scalar, three vector fields, three two-forms and a three-form. However, we will only consider pure gravitational solutions of the eleven dimensional theory, so that all these fields can be set to zero. 2 with T ≡ Tijδ ij , and Li αL α j = δi j , Li αL j βδij = gαβ , Li αL j βg αβ = δij . (7) As usual, curved directions are labeled by greek indices, while flat ones are labeled by latin, and µ, a = 0, 1, . . . , 7 are spacetime coordinates, while α, i = 8, 9, 10 are in the group manifold. Note also that upper indices in the gauge field, Aα µ, are always curved. We will turn on scalars in the diagonal Li α = diag(eλ1 , eλ2 , eλ3) , λ1 + λ2 + λ3 = 0 , (8) and in order to describe the worldvolume of the wrapped D6-branes on the deformed 3-cycle we will choose a metric ansatz of the form2 ds2 8 = α2 1σ 2 1 + α2 2σ 2 2 + α2 3σ 2 3 + e2fds2 1,3 + dρ2 . (9) All four functions αi, f as well as the scalars λi and the dilaton Φ depend only on ρ, and the left-invariant Maurer–Cartan SU(2) 1-forms satisfy dσi = 1 2 ǫijkσj ∧ σk . (10) In this basis we also expand the gauge field 1-forms as Aα = Aα i σ i, with components Aα i that depend only on the variable ρ. For simplicity, the four-dimensional metric ds2 1,3 will be taken to be the Minkowski metric, but our analysis can be easily extended to Ricci-flat metrics. We will represent the 32 × 32 gamma matrices in 11 dimensions as Γa = γa × 12, Γ̂i = γ9 × τ i , (11) where the γa’s denote the 16 × 16 gamma matrices in 8 dimensions and as usual γ9 = iγ0γ1 . . . γ7, so that γ2 9 = 1. Also τ i are Pauli matrices. It will prove useful to introduce Γ9 ≡ 1 6i ǫijkΓ̂ ijk = −iΓ̂1Γ̂2Γ̂3 = γ9 × 12. Within this ansatz, the only consistent way to obtain non-trivial solutions to the Killing spinor equations is to impose on the spinor ǫ the projections Γijǫ = −Γ̂ijǫ , Γ7ǫ = −iΓ9ǫ . (12) 2We should note that deformation of the 3-cycle requires the existence of non-trivial scalars on the coset manifold. 3 The first of these projections relates the two SU(2) algebras obeyed separately by the sets of generators {Γij} and {Γ̂ij} and consequently the “spacetime” and internal deformed 3-spheres. It also states that only singlets of the diagonal SU(2)D of the tensor product of the two SU(2)’s are allowed. We emphasize that simple algebraic considerations reveal that the only allowed coefficient in a relation of the form Γijǫ = λ Γ̂ijǫ is λ = −1. We also note that among the possible pairs {ij} = {12, 23, 31} only two are independent. Therefore the projections (12) represent three conditions in total, thus reducing the number of supersymmetries to 32/23 = 4.3 In the forthcoming derivation of the equations, the relations ΓiΓ̂jǫ = Γ̂iΓjǫ , i 6= j , Γ1Γ̂1ǫ = Γ2Γ̂2ǫ = Γ3Γ̂3ǫ , (13) which can be readily derived from (12), will also be useful. When we wrap the D6-branes on the 3-cycle the SO(1, 6) × SO(3)R symmetry group of the unwrapped branes is broken to SO(1, 3) × SO(3) × SO(3)R. The worldvolume of the brane will support covariantly constant spinors after some twisting or mixing of the spin and gauge connections. In the presence of scalars this twisting can not be simply performed through a direct identification of the spin and gauge connections. As detailed in the appendix the gauge field is defined through the generalized twist4 1 g ω23 1 α1 + A1 1 α1 cosh λ23 + A2 2 α2 sinhλ31 − A3 3 α3 sinhλ12 = 0 (14) and cyclic in 1, 2, 3, so that the solution is A1 1 = α1 g [ − ω23 1 α1 cosh λ23 + eλ21 sinhλ31 ω31 2 α2 − eλ31 sinhλ12 ω12 3 α3 ] , (15) and so on for A2 2 and A3 3. We have used the notation ωjk i = ǫijk α2 j + α2 k − α2 i 2αjαk , (16) 3All conditions in (12) can be cast in the form (Γ7Γ̂i + 1 2 ǫijkΓjk)ǫ = 0 . 4In the absence of scalar fields, with Li α = δi α, and with no deformation of the 3-sphere, the gauge field reduces simply to Ai i = − 1 2g [37]. 4 for the components of the spin connection along the 3-sphere expanded as ωjk = ωjk i σi, and λij = λi − λj. We see that, in general, the relation between the spin connection and the gauge field involves in a rather complicated way the scalar fields. A detailed account of the computations required to derive the equations obeyed by the various fields is given in the appendix. Here we just collect the results. From the gaugino variation in (2) one obtains the equations obeyed by the dilaton dΦ dρ = 1 2 eΦ ( eλ1 α2α3 F 1 23 + eλ2 α3α1 F 2 31 + eλ3 α1α2 F 3 12 ) + g 8 e−Φ(e2λ1 + e2λ2 + e2λ3) . (17) and by the scalars, dλ1 dρ = eΦ 3 ( 2 eλ1 α2α3 F 1 23 − eλ2 α3α1 F 2 31 − eλ3 α1α2 F 3 12 ) − g 6 e−Φ(2e2λ1 − e2λ2 − e2λ3) , (18) and cyclic in the 1, 2, 3 indices for the other two equations. Also we have denoted the field strength components by F i jk in the σj ∧ σk basis. In terms of the gauge field components they read F 1 23 = A1 1 + gA2 2A 3 3 , and cyclic permutations . (19) From the gravitino equation one determines the warp factor f in terms of the dilaton Φ as f = Φ 3 , (20) as well as the differential equation 1 α1 dα1 dρ = eΦ 6 ( eλ1 F 1 23 α2α3 − 5eλ2 F 2 31 α3α1 − 5eλ3 F 3 12 α1α2 ) + g 24 e−Φ(e2λ1 + e2λ2 + e2λ3) , (21) together with two more equations obtained by cyclic permutations of the 1, 2, 3 indices. Furthermore, from δψρ we can obtain the radial dependence of the spinor ǫ, which is simply given by ǫ = ef/2ǫ0 = eΦ/6ǫ0 , (22) for ǫ0 a constant spinor obeying the projection conditions (12). This radial dependence is of the general form ǫ = g 1/4 00 ǫ0, which can be proved using general arguments based on the supersymmetric algebra. The dependence on the particular model is only via the projections imposed on the constant spinor ǫ0, which reduce the number of its independent components (see for instance [51]). Using the appropriate formulae in [50] we may lift our 8-dimensional background into a full solution of 11-dimensional supergravity with only the metric turned on. The result 5 is of the form ds2 11 = ds2 1,3 + ds2 7 where the 7-dimensional part is ds2 7 = e−2Φ/3dρ2 + e−2Φ/3 3 ∑ i=1 α2 iσ 2 i + e4Φ/3 3 ∑ i=1 e2λi(2/gΣi + 2Ai iσi) 2 . (23) This metric, when the various functions are subject to the conditions (15) and (17)-(21), describes G2 holonomy manifolds with an SU(2) × SU(2) isometry. It is worth examining what the Killing spinor in (22) represents from an eleven dimen- sional point of view. Recall that, in general, when a supersymmetry variation parameter ǫ is lifted from eight to eleven dimensions, it is multiplied by a factor, i.e. ǫ11 = e−Φ/6ǫ.5 Using, in our case, the expression (22) we see that the constant spinor ǫ0 is indeed the 11-dimensional spinor which, being subject to the projections (12), has 4 independent components. We will next show that it splits into the form ǫ0 = ǫ1,3 × ǫ7 in such a way that the spinor ǫ7 in seven dimensions has only one independent component, in agreement with the correct amount of independent supercharges preserved by a G2 holonomy mani- fold. In order to proceed we specialize the index µ to represent only the flat directions, i.e. µ = 0, 1, 2, 3 and we denote by µ̄ = 4, 5, 6, 7 the rest. Then we may represent the gamma matrices in 11-dimensions as Γµ = γµ × 14 × 12 , Γµ̄ = γ5 × γµ̄ × 12 , (24) Γ̂i = γ5 × γ̄5 × τi , where γ5 = iγ0γ1γ2γ3, γ̄5 = γ4γ5γ6γ7 and where we have used that ǫ0123 = ǫ4567 = 1. Using the split ǫ0 = ǫ1,3 × ǫ7 we see that the projections (12) imply (γij × 12)ǫ7 = −(14 × τij)ǫ7 , (γ7 × 12)ǫ7 = −i(γ̄5 × 12)ǫ7 . (25) These are 8 conditions in total on the 8-component spinor ǫ7 and therefore the latter has indeed only one independent component, as advertised. Moreover, as shown in footnote 6 below the spinor ǫ7 is G2 invariant. The spinor ǫ1,3 is subject to no conditions at all and therefore the N = 1 supersymmetry in four dimensions is intact we may have reduced supersymmetry if the Minkowski space is replaced by a Ricci flat manifold which admits less Killing spinors that Minkowski space). 5This corrects an apparent typo in equation (34) of [50]. 6 For completeness we also construct the 3-form which is closed and co-closed and whose existence implies that the manifold has G2 holonomy. On general grounds its components in the 7-bein basis ea∧eb∧ec are of the form Φ (3) abc = iǭ7Γabcǫ7, where a, b, c = 1, 2, . . . , 7 and the gamma matrices in seven dimensions are the corresponding part of the decomposition (24). Using the split a = (i, î, 7), where i = 1, 2, 3 and î = i+3, as well as the normalization choice iǭ7Γ123ǫ7 = 1, we find that Φ(3) = 1 6 ψabc e a ∧ eb ∧ ec , (26) where ψabc are the octonionic structure constants with non-vanishing components in our basis being given by6 ψijk = ǫijk , ψiĵk̂ = −ǫijk , ψ7iĵ = δij . (27) It is convenient to cast the metric and the equations in the different form ds2 7 = dr2 + 3 ∑ i=1 a2 iσ 2 i + 3 ∑ i=1 b2i (Σi + ciσi) 2 , (28) where ci = 2Ai i and ai = e−Φ/3αi , bi = e2Φ/3eλi , e2Φ = b1b2b3 , dr = e−Φ/3dρ . (29) Then the equations (17), (18) and (21) become da1 dr = − b2 a3 F 2 31 − b3 a2 F 3 12 , db1 dr = b21 a2a3 F 1 23 − g 4b2b3 (b21 − b22 − b23) , (30) and cyclic in the 1, 2, 3 indices, where the field strength components in (19) are computed using A1 1 = a1 g [ − a2 2 + a2 3 − a2 1 2a1a2a3 b22 + b23 2b2b3 + b23 − b21 2b3b1 b2 b1 a2 3 + a2 1 − a2 2 2a1a2a3 − b21 − b22 2b1b2 b3 b1 a2 1 + a2 2 − a2 3 2a1a2a3 ] = − 1 g d2 2 + d2 3 − d2 1 2d2d3 ≡ − 1 g Ω23 1 , (31) 6In the same basis the non-vanishing components of the G2 invariant 4-index tensor ψabcd are ψ 7ijk̂ = ǫijk , ψ 7̂iĵk̂ = −ǫijk , ψijm̂n̂ = δimδjn − δinδjm . Using these, one can show that the projectors (25) imply that (Γab + 1 4 ψabcdΓcd)ǫ7 = 0 , which is precisely the condition for a G2 invariant Killing spinor. 7 where di ≡ ai bi and cyclic in 1, 2, 3. We see that the generalized twist condition (15) takes the form of the ordinary twist, but for an auxiliary 3-sphere deformed metric obtained by replacing the ai’s in the metric (28) by the di’s defined above. In the rest of this paper we will set the parameter g = 2 which is equivalent to the rescaling bi → big/2. This does not apply for the various formulae in the appendix. It is worth examining the limit where the radius of the “spacetime” 3-sphere becomes very large so that it can be approximated by IR3. This means that effectively the D6-branes are unwrapped. This limit can be taken systematically as follows: consider the rescaling σi → ǫdxi, bi → ǫbi and r → ǫr in the limit ǫ → 0. Then, since the functions ci = 2Ai i do not scale, the metric (28) takes the form ds2 7 = dx2 i + ds2 4, where the four-dimensional non-trivial part of the metric is ds2 4 = dr2 + 3 ∑ i=1 b2i Σ 2 i . (32) The coefficients bi as functions of r obey a set of differential equations that also follow from the above mentioned limiting procedure from (30). Indeed the first equation in (30) reduces to the statement that the coefficients ai = constant and therefore they can be absorbed into a rescaling of the new coordinates xi, as we have already done above. The result is db1 dr = 1 2b2b3 (b22 + b23 − b21) , and cyclic permutations . (33) This is nothing but the Lagrange system or, equivalently, the Euclidean version of the Euler spinning top system. The four-dimensional metrics (32) governed by that system correspond to a class of hyperkähler metrics with SU(2) isometry with famous example, when an extra U(1) symmetry develops (i.e., for instance when b2 = b3), the Eguchi– Hanson metric which is the first non-trivial ALE four-manifold.7 This is in agreement with the fact that the near horizon limit of D6-branes of type IIA when uplifted to M- theory, contains, besides the D6-brane worldvolume, the Eguchi–Hanson metric. Returning back to the generic case, it is obvious that integrating the system of first order non-linear equations (30) is a difficult task in general. Nevertheless one can show 7In fact, the Eguchi–Hanson metric is the only regular metric in the family described by (33). As it was shown in [52] a generalization of it with b1 6= b2 6= b3 6= b1 leads to singular metrics. It can be shown that, from a string theoretical view point, this corresponds to continuous distributions of D6-branes in type IIA with physically unacceptable densities. 8 that I = a1a2a3 − a1b2b3c2c3 − a2b3b1c3c1 − a3b1b2c1c2 , (34) is a constant of motion. The existence of this constant of motion fits well with the fact that the 3-form in (26), after using the explicit basis (37) below in terms of the SU(2) Maurer–Cartan 1-forms, can be written as Φ(3) = Iσ1 ∧ σ2 ∧ σ3 + dΛ , (35) where I is the conserved quantity in (34) and Λ is some 2-form. Hence the conservation of I is a direct consequence of the closure of the 3-form Φ(3), and appears as the coefficient of the volume form of the “spacetime” 3-sphere. Notice that there is no conserved quantity associated with the internal 3-sphere.8 A promising avenue towards finding explicit new solutions will arise if the system (30) can be related to well studied in the literature spinning top-like systems which in many cases are integrable. This is the line of approach advocated in [53], but will leave this and related investigations for future research. Let us now consider the consistent truncation a2 = a3 and b2 = b3, where an extra U(1) symmetry develops. Then after some algebra we conclude that the remaining four independent functions obey the system9 ȧ1 = 1 4 a3 1b 4 2 a4 2b 3 1 , ȧ2 = 1 2 b1 a2 − 3 8 a2 1b 2 2 a3 2b1 + 1 8 a2 1b 4 2 a3 2b 3 1 , ḃ1 = − 1 2 b21 a2 2 + 3 8 a2 1b 2 2 a4 2 − 1 2 ( b21 b22 − 2 ) , (36) ḃ2 = 1 2 b1 b2 − 1 8 a2 1b 5 2 a4 2b 3 1 , 8In the notation of [28] p = I and q = 0. In principle, the information contained into our system (30) for the metric (28) is also encoded into equations (80)-(81) of [28] for the metric (78)-(79) of the same reference. These equations are highly non-linear second order equations for three functions. In our approach they would arise upon eliminating three among our six unknown functions. A simple counting argument shows that in both cases the number of integration constants is the same. We note here that it does not seem possible to investigate metrics with both p 6= 0 and q 6= 0 using eight-dimensional gauged supergravity. The reason is that, in the original metric ansatz (9) there cannot be by definition any dependence on the internal SU(2) coordinates that parameterize the Maurer–Cartan 1-forms Σi. 9It is straightforward to verify that the further consistent truncation with a1 = a2 = a3 and b1 = b2 = b3 gives a system which is trivially solved, leading to the metric of [49]. 9 where we have used that in this case c2 = − a1b2 2a2b1 and c1 = 2c22 − 1. This system coincides (after we let r → −r) with that in equation (23) of [29] and in the limit of a1 = 0 it is just the system corresponding to the resolved conifold. We will finally show how the system of equations (30) can also be derived from self- duality of the spin connection for the seven manifold. In order to do so, we will split the indices in (28) as before, namely as a = (i, î, 7), and use the 7-bein basis e7 = dr , ei = aiσi , eî = bi(Σi + ciσi) , i = 1, 2, 3 , î = i+ 3 . (37) We then compute dei = ȧi ai e7 ∧ ei + 1 2 ai ajak ǫijk e j ∧ ek , deî = ḃi bi e7 ∧ eî + biċi ai e7 ∧ ei + 1 2 bi ǫijk ( 1 bjbk eĵ ∧ ek̂ + ci + cjck ajak ej ∧ ek − 2 cj ajbk ej ∧ ek̂ ) , (38) where the dot stands for d dr . Using then the Cartan’s structure equations dea +ωab∧eb = 0 we compute the spin connection ωi7 = ȧi ai ei + biċi 2ai eî , ω î7 = ḃi bi eî + biċi 2ai ei , ωij = 1 2 ǫijk ( ai ajak + aj aiak − ak aiaj ) ek − 1 2 ǫijk bk aiaj (ck + cicj) e k̂ , (39) ω îĵ = 1 2 ǫijk ( bi bjbk + bj bibk − bk bibj ) ek̂ − 1 2 ǫijk ck ak ( bi bj + bj bi ) ek , ωiĵ = biċi 2ai δij e 7 + 1 2 ǫijk bj aiak (cj + ckci) e k − 1 2 ǫijk ci ai ( bj bk − bk bj ) ek̂ . Then, let us recall that imposing the self-duality condition on the spin connection, i.e. ωab = 1 2 ψabcdωcd, where ψabcd is the G2 invariant 4-index tensor, is equivalent in our basis to the following seven equations ω7i = ǫijkω jk̂ , ω7̂i = 1 2 ǫijk(ω jk − ωĵk̂) , (40) ω îi = 0 . 10 Applying these to our case we obtain the differential equations (30) and the generalized twist condition (31), plus the condition ∑3 i=1 bi ai ċi = 0, which is equivalent to (A.11) in the appendix and is satisfied automatically once (30) and (31) are. Since self-duality of the spin connection in seven dimensions implies that the 3-form defined in (26) is closed and co-closed and, therefore, G2 holonomy (noted in [54, 55], proved explicitly in [25] and used to rederive the metric of [49] in [56]) we have shown that our equations (30) (or equivalently (17), (18) and (21)) indeed describe a manifold of G2 holonomy. It will be interesting to extend the eight-dimensional gauged supergravity approach to G2 manifolds in order to find general conditions for manifolds with weak G2 holonomy [57] having an SU(2)×SU(2) isometry. The main difference in this case is that the three form is no longer closed, i.e. it obeys dΦ(3) ∼ ∗Φ(3) and consequently the Minkowski metric ds2 1,3 has to be replaced by an Einstein space with negative cosmological constant. Nevertheless, supersymmetry can be preserved and a generalization of the self-duality condition on the spin connection (40) leading to manifolds with weak G2 holonomy also exists [25]. We also believe that the eight-dimensional approach to G2 manifolds will also prove useful in the investigation of Spin(7) manifolds. We hope to report work along these lines in the future. Acknowledgments R. H. acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131 Quantum Spacetime and by the Swiss Office for Education and Science and the Swiss National Science Foun- dation, and hospitality of the Erwin Schrödinger Institute in Vienna. K. S. acknowledges the hospitality and the financial support of the Institute of Physics at the University of Neuchâtel in which he was a member while part of this work was done. A Appendix In this appendix we will provide some details on the derivation of the Killing spinor equations and conditions (14)-(22). As already noted, the only consistent way to obtain a consistent set of differential equations from the supersymmetry variations (2) and (3) is to impose projections (12) on the spinors. We also provide for convenience the expressions for Pµ ij and Qµ ij defined in 11 (5). For the diagonal matrix Li α in (8), it is convenient to represent them as forms in the index µ, Pij =   ∂λ1 gA3 sinhλ12 gA2 sinhλ31 gA3 sinh λ12 ∂λ2 gA1 sinhλ23 gA2 sinh λ31 gA1 sinhλ23 ∂λ3   (A.1) and Qij =   0 −gA3 cosh λ12 gA2 cosh λ31 gA3 cosh λ12 0 −gA1 coshλ23 −gA2 coshλ31 gA1 coshλ23 0   . (A.2) We start with the i = 1 case in the gaugino equation, δχ1 = 0, which implies two different equations: factorizing Γ̂2Γ̂3 ǫ we get dλ1 dρ + 2 3 dΦ dρ = eΦ+λ1 F 1 23 α2α3 − g 4 e−Φ(e2λ1 − e2λ2 − e2λ3) , (A.3) where F 1 23 is defined in (19). In addition, from terms proportional to Γ̂2Γ3 ǫ we get eΦ+λ1 F 1 ρ1 α1 + g ( A3 3 α3 sinh λ12 − A2 2 α2 sinh λ31 ) = 0 , (A.4) where F 1 ρ1 = ∂ρA 1 1, or equivalently, after the change of variables (29) (and setting g = 2) b1ċ1 a1 + c3 a3 ( b1 b2 − b2 b1 ) − c2 a2 ( b3 b1 − b1 b3 ) = 0 . (A.5) The four additional equations corresponding to δχi = 0, for i = 2, 3 can be obtained from (A.3) and (A.4) by cyclic permutations in the indices 1, 2, 3. Using then the constraint λ1 + λ2 + λ3 = 0 we get dλ1 dρ = eΦ 3 ( 2 eλ1 α2α3 F 1 23 − eλ2 α3α1 F 2 31 − eλ3 α1α2 F 3 12 ) − g 6 e−Φ(2e2λ1 − e2λ2 − e2λ3) (A.6) and dΦ dρ = 1 2 eΦ ( eλ1 α2α3 F 1 23 + eλ2 α3α1 F 2 31 + eλ3 α1α2 F 3 12 ) + g 8 e−Φ(e2λ1 + e2λ2 + e2λ3) . (A.7) Next we turn to the gravitino variation, δψµ = 0, where we should distinguish two possibilities, according to whether µ is a coordinate on the wrapped 3-sphere or on the unwrapped directions. Thus, if µ = σ1, from Γ̂2Γ̂3 ǫ we get ω23 1 + gA1 1 cosh λ23 − α1 eΦ 6 ( eλ2 α2 F 2 ρ2 + eλ3 α3 F 3 ρ3 − 5 eλ1 α1 F 1 ρ1 ) = 0 (A.8) and, from terms proportional to Γ̂2Γ3 ǫ 1 α1 dα1 dρ − 1 6 eΦ ( eλ1 α2α3 F 1 23 − 5 eλ2 α3α1 F 2 31 − 5 eλ3 α1α2 F 3 12 ) − g 24 e−Φ(e2λ1 +e2λ2 +e2λ3) = 0 , (A.9) 12 where we have used the spin connection component ωiρ i = dαi dρ for the metric ansatz (9). The equations obtained from cyclicity of (A.8) and (A.9) correspond to the choices µ = σ2, σ3. The generalized twist (14) is then derived from (A.4) and (A.8). It amounts to turning on a gauge field given by (15). It is important to verify that substituting back (15) into (A.4) and after using (A.6), (A.7) and (A.9), gives no new constraints for the various functions. After a straightforward but lengthly computation, one can show that this is indeed the case. 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